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Differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiation4 Marks
Question
If $(\text{x}-\text{y})\text{e}^{\frac{\text{x}}{\text{x}-\text{y}}}=\text{a},$ prove that $\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}=2\text{y}$
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Answer

Consider the given function, $(\text{x}-\text{y})\text{e}^{\frac{\text{x}}{\text{x}-\text{y}}}=\text{a}.$
We need to prove that $\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}=2\text{y}.$
Differentiating the given equation w.r.t 'x' we get
$(\text{x}-\text{y})\Bigg[\text{e}^{\frac{\text{x}}{\text{x}-\text{y}}}\Bigg(\frac{(\text{x}-\text{y})-\text{x}\big(1-\frac{\text{dy}}{\text{dx}}\big)}{(\text{x}-\text{y})^2}\Bigg)\Bigg]+\text{e}^\frac{\text{x}}{\text{x}-\text{y}}\Big(1-\frac{\text{dy}}{\text{dx}}\Big)=0$
$\Rightarrow\frac{(\text{x}-\text{y})-\text{x}\Big(1-\frac{\text{dy}}{\text{dx}}\Big)}{(\text{x}-\text{y})}+\Big(1-\frac{\text{dy}}{\text{dx}}\Big)=0$
$\Rightarrow\Big(1+\frac{\text{dy}}{\text{dx}}\Big)\Big(1-\frac{\text{x}}{\text{x}-\text{y}}\Big)+1=0$
$\Rightarrow\Big(1+\frac{\text{dy}}{\text{dx}}\Big)\Big(\frac{-\text{y}}{\text{x}-\text{y}}\Big)+1=0$
$\Rightarrow-\text{y}+\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}-\text{y}=0$
$\Rightarrow\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}=2\text{y}$

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